国际经济学专题2：The Heterogeneous Impact of Market Size on Innovation

本专栏是笔者2025年春季学习国际经济学专题课程的一些笔记，由于笔者并未学过一些前置知识（或者学的很差），内容存在很多繁琐累赘之处，同时可能存在错误或片面理解，请见谅

这篇blog的内容来源于：Philippe Aghion, Antonin Bergeaud, Matthieu Lequien, Marc J. Melitz; The Heterogeneous Impact of Market Size on Innovation: Evidence from French Firm-Level Exports. The Review of Economics and Statistics 2024; 106 (3): 608–626.
链接：MIT Press
本blog中所有图片均来源于论文原图

1. Background

Key point: Skewed innovation response to common demand shocks arises

• “Standard” market size effect: Increase innovation for all firms
• Endogenous competition effect: Discourages innovation by low productivity firms

Why Skewed:

• The expanded market for exports will attract new firms into the export market as more firms find it profitable to sell their products there;
• This in turn will raise competition for exporters into that market. Due to the nature of competition between firms – featuring endogenous markups – this effect gradually dissipates as productivity (and resulting market share) increases.

2. Evidence

Evidence of an induced competition effect: The correlation between a local demand shock and measures of ensuing product entry into that destination

• Demand shock: Exports from all countries excluding France (The same as previous parts)
• Net entry rate: Dividing the net change in the number of products sold by the number of existing products

3. Basic setup

3.1 Utility

denote the number of consumers (Market Size) in Export market destination . is the measure of available products

Sub-Utility (Marshall’s Second Law of Demand): where , and

3.2 Consumer optimization

Representative consumer facing prices : Lagrange multiplier: Inverse residual demand function:

Higher shifts all residual demand curves downwards, as an increase in competition for a given exogenous level of market size .

3.3 Firm optimization

Firm with marginal cost facing competition

Maximize operating profits F.O.C.

Firms with high cost() do not produce

The maximized profit per consumer:

Output and profit are decreasing in both firm level cost and the endogenous competition measure .

• More productive firms (with lower cost ) are larger (with more output) and earn higher profits than their less productive counterparts
• An increase in competition lowers production levels and profits for all firms.

4. Innovation choice

A firm can reduce its marginal cost of production below its baseline cost by investing in innovation. Let is the firm’s investment in innovation: Firm choose its optimal R&D investment so as to maximize total profit (Assume that the cost of innovation is quadratic in ): F.O.C. Total firm output (across consumers):

Additional condition 1: Minimal baseline cost

• The post-innovation marginal cost is bounded away from zero, even for the most productive firms.

Additional condition 1: Smooth innovation response

• S.O.C. The slope of the marginal cost is strictly larger than the slope of the marginal gain

Considering the marginal cost & gain of innovation All firms face the same marginal cost curve and their marginal gain curves have the same slope

• The lower firms have a higher intercept, thus a higher marginal gain, and therefore invest more in R&D
• Firms with sufficiently high baseline costs do not innovate.

5. Market size and competition effects

5.1 Market size effect

Analyze the direct effect of an increase in , holding the competition level constant first.

Innovation response for firms with different baseline costs - Same marginal cost curve - Marginal gain curves change - The slope increases but it's still the same - The intercept of low baseline cost firm increases more

This increase in market size also induces some firms to begin R&D.

5.2 Competition effect

Consider the effect of an increase in competition , holding market size constant.

Innovation response for firms with different baseline costs - Same marginal cost curve - Marginal gain curves change - The slope also increases but it's still the same - The intercept of low baseline cost firm decreases more

The new marginal benefit curve remains below the old one at least until it meets the marginal cost curve, even though an increase in competition increases the slope of the marginal benefit curve.

The competition increase also induces some firms to stop R&D

5.3 Conclusion

An increase in market size will have two effects on firms’ innovation incentives:

• A direct-positive-market size effect
  • The increase in induces all firms to increase innovation;
  • This effect was shown above to be more positive for more frontier firms (i.e. for firms with lower initial production cost );
• An induced-negative-competition effect
  • The increase in increases competition which in turns reduces firms’ innovation incentives;
  • This effect of an increase in on firms’ innovation is more negative for less productive firms (i.e. for firms with higher initial production cost ).

The overall effect of an increase in market size on innovation, which combines the direct market size effect and the induced competition effect, will be unambiguously more positive for more frontier firms;

This overall effect can turn out to be negative for the least productive firms -- depending on the relative magnitude of the direct and indirect impacts.

This heterogeneous response is fully consistent with the empirical analysis:

• the most productive half of the firms increase their innovation when their market size expands
• while the response for the least productive half of the firms is essentially muted.

Theoretical appendix

Competition level in destination increases with

Although this equilibrium involves all the firms operating in D, including both the exporters to along with the domestic producers in , the equilibrium competition level is determined independently of the export supply to (which then only impacts the number of domestic entrants and producers).

Like The Impact of Trade On Intra-industry Reallocations and Aggregate Industry Productivity (Melitz 2003), we have:

Zero Profit Condition:

where denote the fixed production cost faced by those domestic firms in . Only domestic firms with a baseline cost below a cutoff value is profitable.

Free-Entry Condition:

where denote the sunk entry cost, and denote the cumulative distribution of baseline costs among domestic producers in .

Proposition 1 The two conditions and jointly determine a unique pair .

Proposition 2 An increase in market size in leads to an increase in competition .

The Proof is detailed in the appendix of Aghion et al. (2024)

Proposition 3 For a given innovating firm, innovation, output and output holding fixed all move comonotonically in response to a change in the market size .

Proposition 4 (a) For any cost distributions , there exists values of and such that some high cost firms reduce their innovation when the market size L increases; (b) For any values of and , there exists cost distributions and a range of such that some high cost rms reduce their innovation when the market size L increases within that range.

The Proof of Proposition 3&4 is detailed in the appendix of Aghion et al. (2018)